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This paper deals with the observer design of nonlinear tubular chemical reactor models. The analysis is performed on a linearized version of the model around a steady-state profile, in which some coefficients are a function of the spatial variable. The study starts from a general model of chemical tubular reactors that will serve as a benchmark model for the formulation in the infinite dimensional Hilbert state space. The optimal output injection operator associated to the proposed observer is computed via the solution of the infinite-dimensional Riccati equation in the space variable. The performance of the observer is illustrated through numerical experiments of an industrial pulp bleaching tubular reactor model.